Start by activating the yeast in the warm water, stirring until it's fully dissolved. Let the mixture sit for about 10 minutes until bubbles form.
In a stand mixer bowl, combine the flour, salt, sugar, and olive oil. Then, add in the yeast and water mixture. Using the dough hook attachment, knead the dough until all ingredients are incorporated. If needed, add a tablespoon or two of water and continue kneading until you have a smooth, elastic dough.
Drizzle the extra olive oil over the dough, ensuring it's coated all around. Cover the bowl with plastic wrap or a towel and let it rise in a warm area until it doubles in size, which should take around an hour.
Once the dough has risen, transfer it to a lightly floured surface and divide it into 6-8 equal portions. Shape these into balls, cover with a towel, and let them rest for another 20-30 minutes.
As the dough rests, get your fillings ready. Mix the zaatar with olive oil for one filling, and combine the two cheeses for the other.
Preheat your oven to 450°F/230°C and line a baking sheet with parchment paper.
Roll each dough ball into a thin, round disc (about 8 inches in diameter). Spread 1-2 tablespoons of the zaatar mixture on half of the discs, and sprinkle the cheese mixture over the other half.
Bake the flatbreads in the preheated oven for about 10 minutes, or until they're fully cooked. If you're using a pizza oven, it will take approximately 2 minutes to cook. Serve the flatbreads while they're still warm.